December 2012

Problem 655 | Beam Deflection by Conjugate Beam Method

Problem 655
Find the value of EIδ under each concentrated load of the beam shown in Fig. P-655.
 

655-conjugate-beam-method.gif

 

Problem 656 | Beam Deflection by Conjugate Beam Method

Problem 656
Find the value of EIδ at the point of application of the 200 N·m couple in Fig. P-656.
 

656-conjugate-beam-method.gif

 

Problem 657 | Beam Deflection by Conjugate Beam Method

Problem 657
Determine the midspan value of EIδ for the beam shown in Fig. P-657.
 

657-conjugate-beam-method.gif

 

Problem 658 | Beam Deflection by Conjugate Beam Method

Problem 658
For the beam shown in Fig. P-658, find the value of EIδ at the point of application of the couple.
 

658-conjugate-beam-method.gif

 

Restrained Beams

Something is Strange About the Vertical Reactions of Propped Beams

Restrained Beams
In addition to the equations of static equilibrium, relations from the geometry of elastic curve are essential to the study of indeterminate beams. Such relations can be obtained from the study of deflection and rotation of beam. This section will focus on two types of indeterminate beams; the propped beams and the fully restrained beams.
 

Application of Double Integration and Superposition Methods to Restrained Beams

Superposition Method

There are 12 cases listed in the method of superposition for beam deflection.

  • Cantilever beam with...
    1. concentrated load at the free end.
    2. concentrated load anywhere on the beam.
    3. uniform load over the entire span.
    4. triangular load with zero at the free end
    5. moment load at the free end.
  • Simply supported beam with...
    1. concentrated load at the midspan.
    2. concentrated load anywhere on the beam span.
    3. uniform load over the entire span.
    4. triangular load which is zero at one end and full at the other end.
    5. triangular load with zero at both ends and full at the midspan.
    6. moment load at the right support.
    7. moment load at the left support.

See beam deflection by superposition method for details.
 

Problem 704 | Solution of Propped Beam

Reactions of Propped Beam by Double Integration Method | Theory of Structures

Problem 704
Find the reactions at the supports and draw the shear and moment diagrams of the propped beam shown in Fig. P-704.
 

704-propped-beam-uniform-load.gif

 

Problem 705 | Solution of Propped Beam with Increasing Load

Problem 705
Find the reaction at the simple support of the propped beam shown in Fig. P-705 and sketch the shear and moment diagrams.
 

Propped beam loaded with triangular or uniformly varying load

 

Continuous Beams

Continuous beams are those that rest over three or more supports, thereby having one or more redundant support reactions.
 

These section includes
1. Generalized form of three-moment equation
2. Factors for three-moment equation
3. Application of the three-moment equation
4. Reactions of continuous beams
5. Shear and moment diagrams of continuous beams
6. Continuous beams with fixed ends
7. Deflection determined by three-moment equation
8. Moment distribution method
 

The Three-Moment Equation

The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.
 

Consider three points on the beam loaded as shown.
 

000-three-moment-equation.gif

 

Problem 706 | Solution of Propped Beam with Decreasing Load

Example 03
The propped beam shown in Fig. P -706 is loaded by decreasing triangular load varying from wo from the simple end to zero at the fixed end. Find the support reactions and sketch the shear and moment diagrams
 

Propped with decreasing load from w at simple support to zero at the fixed end.

 

Problem 707 | Propped Beam with Moment Load

Problem 707
A couple M is applied at the propped end of the beam shown in Fig. P-707. Compute R at the propped end and also the wall restraining moment.
 

707-propped-beam-moment-load.gif

 

Problem 708 | Two Indentical Cantilever Beams

Problem 708
Two identical cantilever beams in contact at their ends support a distributed load over one of them as shown in Fig. P-708. Determine the restraining moment at each wall.
 

Two cantilever beams.

 

Problem 709 | Propped Beam with Spring Support

Example 06
The beam in Figure PB-006 is supported at the left by a spring that deflects 1 inch for each 300 lb. For the beam E = 1.5 × 106 psi and I = 144 in4. Compute the deflection of the spring.
 

Beam with spring support

 

Problem 710 | Two simple beams at 90 degree to each other

Problem 710
Two timber beams are mounted at right angles and in contact with each other at their midpoints. The upper beam A is 2 in wide by 4 in deep and simply supported on an 8-ft span; the lower beam B is 3 in wide by 8 in deep and simply supported on a 10-ft span. At their cross-over point, they jointly support a load P = 2000 lb. Determine the contact force between the beams.
 

Problem 711 | Cantilever beam with free end on top of a simple beam

Problem 711
A cantilever beam BD rests on a simple beam AC as shown in Fig. P-711. Both beams are of the same material and are 3 in wide by 8 in deep. If they jointly carry a load P = 1400 lb, compute the maximum flexural stress developed in the beams.
 

The ends of cantilever beam rests on top of simple beam at the third point.

 

Problem 712 | Propped beam with initial clearance at the roller support

Problem 712
There is a small initial clearance D between the left end of the beam shown in Fig. P-712 and the roller support. Determine the reaction at the roller support after the uniformly distributed load is applied.
 

712-propped-beam-with-clearance.gif

 

Problem 713 | Fully restrained beam with symmetrically placed concentrated loads

Problem 713
Determine the end moment and midspan value of EIδ for the restrained beam shown in Fig. PB-010. (Hint: Because of symmetry, the end shears are equal and the slope is zero at midspan. Let the redundant be the moment at midspan.)
 

713-fully-restrained-beams-symmetrical-point-loads.gif

 

Problem 714 | Triangular load over the entire span of fully restrained beam

Problem 714
Determine the end moments of the restrained beam shown in Fig. P-714.
 

714-restrained-beam-triangular-load.gif

 

Solution
$\delta_A = 0$

$\delta_{triangular\,\,load} - \delta_{fixed\,\,end\,\,moment} - \delta_{reaction\,\,at\,\,A} = 0$
 

Problem 715 | Distributed loads placed symmetrically over fully restrained beam

Problem 12
Determine the moment and maximum EIδ for the restrained beam shown in Fig. RB-012. (Hint: Let the redundants be the shear and moment at the midspan. Also note that the midspan shear is zero.)
 

715-restrained-beam-symmetrical-uniform-loads.gif

 

Application of Area-Moment Method to Restrained Beams

Reactions of Propped Beam with Triangular Load by Area Moment Method | Theory of Structures

See deflection of beam by moment-area method for details.
 

Rotation of beam from A to B

$\theta_{AB} = \dfrac{1}{EI}(\text{Area}_{AB})$

 

Deviation of B from a tangent line through A

$t_{B/A} = \dfrac{1}{EI} (Area_{AB}) \, \bar{X}_B$

 

Problem 704 | Propped beam with some uniform load by moment-area method

Propped Beam with Uniform Load by Area Moment Method | Theory of Structures

Problem 704
Find the reaction at the simple support of the propped beam shown in Figure PB-001 by using moment-area method.
 

704-propped-beam-uniform-load.gif

 

Problem 720 | Propped beam with increasing load by moment-area method

Reactions of Propped Beam with Triangular Load by Area Moment Method | Theory of Structures

Problem 720
Find the reaction at the simple support of the propped beam shown in Fig. P-705 by using moment-area method.
 

Propped beam loaded with triangular or uniformly varying load

 

Problem 721 | Propped beam with decreasing load by moment-area method

Reactions of Propped Beam with Triangular Load by Area Moment Method | Theory of Structures

Problem 721
By the use of moment-are method, determine the magnitude of the reaction force at the left support of the propped beam in Fig. P-706.
 

Propped with decreasing load from w at simple support to zero at the fixed end.

 

Pages